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Fuzzy numbers are fuzzy subsets of the set of real numbers satisfying some additional conditions. Fuzzy numbers allow us to model very difficult uncertainties in a very easy way. Arithmetic operations on fuzzy numbers have also been developed, and are based mainly on the crucial Extension Principle. When operating with fuzzy numbers, the results of our calculations strongly depend on the shape of the membership functions of these numbers. Logically, less regular membership functions may lead to very complicated calculi. Moreover, fuzzy numbers with a simpler shape of membership functions often have more intuitive and more natural interpretations. But not only must we apply the concept and the use of fuzzy sets, and its particular case of fuzzy number, but also the new and interesting mathematical construct designed by Fuzzy Complex Numbers, which is much more than a correlate of Complex Numbers in Mathematical Analysis. The selected perspective attempts here that of advancing through axiomatic descriptions.

Historically, the extension of the set of numbers were developed from natural numbers to integers, then rational numbers, real numbers, and finally—in recent times—complex numbers. Similarly, the concept of set has been extended in a variety of ways. A fuzzy set is one such an extension of conventional (classical or crisp) sets. From a point of view of the extension of ranges of characteristic functions, that is, extensions from the discrete set {0,1} to the closed unit interval, [0,1], it looks like the extension from natural numbers to real numbers.

Complex fuzzy sets, which are proposed by Ramot

It is possible to demonstrate that FL is a well-developed logical theory. It includes, amongst others, the theory of functional systems in FL, giving an explanation of what and how it can be represented by formulas of FL calculi. A more general interpretation of FL within other proper categories of fuzzy sets is also feasible (see the book [

Fuzzy logic provides a systematic tool to incorporate human experience. It is based on three core concepts, namely, fuzzy sets, linguistic variables, and possibility distributions. Fuzzy set is used to characterize linguistic variables whose values can be described qualitatively using a linguistic expression and quantitatively using a membership function. Linguistic expressions are useful for communicating concepts and knowledge with human beings, whereas membership functions are useful for processing numeric input data. When a fuzzy set is assigned to a linguistic variable, it imposes an elastic constraint, called a possibility distribution, on the possible values of the variable.

Fuzzy logic is a very rigorous and mathematical discipline of increasing interest [

It is well established that Propositional Logic is isomorphic to Set Theory under the appropriate correspondence between relative components of these two mathematical systems. Furthermore, both of these systems are isomorphic to a Boolean Algebra, which is a well-known mathematical system, defined by abstract entities and their axiomatic properties. The isomorphism between Boolean Algebra, Propositional Logic, and Set Theory guarantees that every theorem in any one of these theories has a counterpart in each of the other two theories. This isomorphism allows us to cover all these theories by developing only one of them. Consequently, we will not spend a lot of time reviewing Mathematical or Crisp Logic; but we spend some time on it in order to reach the comparable concept in Fuzzy Logic.

Fuzzy Complex Sets are given by a complex degree of membership, represented in polar coordinates, which is a combination of a degree of membership in a fuzzy set along with a crisp phase value that denotes the position within the set. The compound value carries more information than a traditional fuzzy set and enables efficient reasoning. We may present a new and generalized interpretation of a complex grade of membership, where a complex membership grade defines a complex fuzzy class. The new definition provides rich semantics that is not readily available through traditional fuzzy sets or complex fuzzy sets and is not limited to a compound of crisp cyclical data with fuzzy data. Furthermore, the two components of the complex fuzzy class carry fuzzy information [

A complex class is represented either in Cartesian or in polar coordinates where both axes induce fuzzy interpretation. Another novelty of this scheme is that it enables representing an infinite set of fuzzy sets.

This provides a new definition of fuzzy complex fuzzy classes along with the axiomatic definition of basic operations on complex fuzzy classes. In addition, coordinate transformation, as well as an extension from two-dimensional fuzzy classes to n-dimensional fuzzy classes are presented.

In the modern sense, the introduction of Complex Numbers is due to Girolamo Cardano (1501–1572). We can observe this in his book

Its use is not a pleasure, according to Cardano, but a “mental torture”. Later (1572), Rafael Bombelli introduced the symbol

In 1629, the Flemish amateur mathematician Albert Girard denominated complex numbers as “impossible solutions”. Girard was of French origin and had emigrated due to religious prosecution.

It was René Descartes who spoke of “imaginary numbers” in 1637. After him, the use of complex numbers was somewhat generalized among mathematicians, amongst others the family Bernoulli (associated with Basel), Leonhard Euler, John Wallis, Caspar Wessel, Abraham De Moivre, Jean Robert Argand, Augustin Cauchy, Karl Friedrich Gauss, Bernhard Riemann, Caspar Wessel, William Rowan Hamilton, Hermann G. Grassmann, William K. Clifford, Lars V. Ahlfors,

But the origins of this theory is much more arcane. In fact, the notion of complex numbers is intimately related with the attempt to prove the famous

The oldest reference known to square roots of negative numbers may be by Heron of Alexandria, around the year 60 AD, when this great mathematician and inventor was calculating volumes of geometric bodies. Two centuries after the aforementioned Heron of Alexandria, Diophantus (about 275 AD) worked on a simple geometrical problem: Find the sides of a right-angled triangle, known its area and perimeter.

We must mention the attempts to find the roots of an arbitrary polynomial by Al-Khwarizmi (

G. J. Toomer, quoted by B. L. Van der Waerden [

The methods of algebra known to the Arabs were introduced in Italy by the Latin translation of the algebra of al-Khwarizmi by Gerard of Cremona (1114–1187) and by the work of Leonardo da Pisa (more known as Fibonacci), who lived from 1170 to 1250.

There are indications that C.F. Gauss (1777–1855) had been in possession of the geometric representation of complex numbers since 1796. He introduced the term

Augustin-Louis Cauchy (1789–1857) initiated the complex function theory in an 1814 memoir submitted to the French Académie des Sciences. The term analytic function was not mentioned in his memoir, but the concept is there.

Hermann G. Grassmann (1809–1877) also introduced the related field of Multidimensional Vector Calculus, and in 1848, James Cockle contributed the split-complex numbers. Combining split-complex numbers and quaternion algebra, W. K. Clifford introduced the so-called

Many years later, when generalizing complex numbers, we can mention the so-called Hyper-Complex Numbers, due to W. R. Hamilton (1805–1865). This scientist introduced a new mathematical construct, the so-called quaternions, in 1843.

Other members of this line of research of multi-complex numbers may be cited:

In more recent times, James J. Buckley [

A fuzzy set,

But many times it may be described, instead, by

_{A}: U → L

Assigning a value, A (x) ∈ L, to each element x ∈U,

_{A} (_{x}_{∈U}

But also can be expressed by

Here, both notations, A (_{A }(

Often, the range of A, denoted by the lattice L, coincides with the closed unit interval of the field of real numbers,

Note that we will use the same symbol for both fuzzy set, A, and membership function associated with A, in the same sense. Both couples,

_{A }(

has the same meaning,

“The element x belongs to A with the membership degree, or equivalently the membership function value, equal either to A(_{A}(

This value expresses the degree of truth that the element x belongs to A.

The fuzzy set A is called ^{*}, with

_{A} (^{*}) = 1

A

And it holds

_{A }[_{A} (_{A }(

A

Such that

(1) N is

(2) N^{α}, the

But if all α-cuts are closed intervals then every fuzzy number must be a convex fuzzy set.

Observe that the converse is not necessarily true.

(3) The

Illustrative examples of fuzzy numbers are the well-known types, such as

- the Fuzzy Trapezoidal Numbers,

and

- the

- bell-shaped membership function (Gaussian);

- L-R fuzzy numbers,

A _{F}, verifies:

(1) m_{F} (

(2) m_{F }(

(3) m_{F} (

(4) m_{F} (

(5) m_{F} (

The more usual way to denote a FGN may be

In the particular case when w = 1, we can express the FGN by

_{LR}

Obviously, when the functions L(

In 1999, Chen and Hsieh [

In particular

And

identifying both

We need to introduce a new representational tool for various different reasons, which justify their necessity and convenience, instead of a simple two-element vector representation.

- It will be easiest for calculation.

- It is physically accurate.

- It is very useful in applications, by using complex algebra.

Let

_{A }(_{A} (_{A }(

Observe that _{A} and ɸ_{A} are both real-valued functions, with an important restriction on r_{A} such that

_{A }(

Therefore, we can consider the precedent membership function, denoted by µ_{A} (x), as composed by two factors,

_{A} (

and

_{A} (

As a particular case, we may consider the case of A with a null membership function, µ_{A} (_{A} (_{A }(_{A} (

We then analyze the membership phase component in more detail, with respect to the fuzzy operations, union, intersection, complement, and so on.

Let

_{A∪B }(x) = [r_{A} (_{B} (_{A∪B} (

▸ here is some

And similarly we can also define

_{A∩B }(_{A} (_{B} (_{A∩B} (

◄ in this case being some T-norm operator.

Their respective phases remain without definition until now,

_{A∪B} (_{A∩B} (

To obtain such definitions, it would be convenient to introduce two auxiliary functions,

In both cases, the domain will be the same fuzzy domain product, also sharing its range:

_{u} = R_{j} = {a ∈

when

At least it must hold the following:

(1) u (a, 0) = a

(2) if |b| ≤ |d|, then |u(a, b)| ≤ |u(a, d)|

(3) u (a, b) = u (b, a)

(4) u (a, u (b, d)) = u (u (a, b), d)

The name of such axioms must be:

(1)

(2)

(3)

(4)

In some cases, it will be convenient to dispose of certain additional axioms for u; as may be:

(5) u is a continuous function

(6) |u (a, a)| > |a|

(7) If |a| ≤ |c|, and |b| ≤ |d |, then |u (a, b)| ≤ u (c, d)

Such axioms are so-called

(5)

(6)

(7)

Furthermore [

The

(1) v (a, 0) = a

(2) if |b| ≤ |d| then |v (a, b)| ≤ | v (a, d)|

(3) v (a, b) = v (b, a)

(4) v (a, v (b, d)) = v (v (a, b), d)

The adequate name of such axioms may be

(1)

(2)

(3)

(4)

In some cases it will be convenient to dispose of certain additional axioms for v, as may be

(5)

(6) |v (a, a)| < |a|;

(7) If |a| ≤ |c|, and |b| ≤ |d|, then |v (a, b)| ≤ v (c, d).

They are so-called

(5)

(6)

(7)

To expose the fundamental operations between fuzzy numbers [

The last case may be defined this way, except when

Two examples are either

or

The difference of the same two fuzzy numbers will be either

or

In the first example for multiplication, it produces

In the case of division between both fuzzy numbers of this example, we have

(1)

(2)

(3)

(4)

(5)

(6)

Recall that a

Let ⊥ ∈ {(+), (-), (*), (:)} with the restriction that for any such operations

^{α}, being 0 < α ≤ 1

Then, we obtain this fuzzy number

_{α∈(}_{0, 1]} [A (⊥) B]^{α}

Its generalization is indeed possible, because we dispose of the

_{ z = x (⊥) y }[min {A(x), B(y)}]

We dispose in this case of

So, we can describe the

_{(x, y)} {min [A(x), B (y)]} = MEET (A, B)

Jointly with

_{(x, y)} {min [A(x), B (y)]} = JOIN (A, B)

Therefore a

Hence, we find that

_{F}, MIN, MAX >

is a distributive lattice, being into the _{F}

Expressing adequate definitions and operational formulae in terms of Fuzzy Numbers [

This above introduced notion of FCNs (Fuzzy Complex Numbers, by acronym) is capable of representing and aggregating a variety of inexact knowledge and data in a unified manner [

For these reasons, the Complex Fuzzy Number must be a new and interesting tool in the continuous advance through the generalization of the (until now often too traditional and monotonic) fields of Science. For more complete and interesting information about these topics, please see the references [

Thank you very much to the referees of this article, who have improved it with their wise comments. And obviously, to the editors and staff of MDPI, for their continuous and invaluable help and support.